A new mathematical solution of convection-dispersion equation to describe solute transport in heterogeneous soils

There was a long-time debate about the validation of the convection-dispersion equation (CDE) and its replacement with the well-known convective lognormal transfer function model (CLT) to describe solute transport in a heterogeneous soil with uniformity at the longitudinal water flow direction and nonuniformity at the transverse direction. The objective of this study is to prove that the CDE is valid and almost identical to the CLT. Gamma probability density function (pdf) was initially assumed in this study to describe the distribution of pore-water velocity across capillary tubes in a heterogeneous soil. The capillary bundle model was used to describe solute transport without transverse solute mixing between adjacent tubes. The inverse-gamma function, a new mathematical solution of the CDE differential equation with scale-dependent dispersivity, was initially derived from the capillary bundle model and the gamma pdf. The only difference between the inverse-gamma function and the CLT is that lognormal pdf of pore-water velocity is assumed in the CLT while the two pdfs are close to each other. The inverse-gamma function and the CLT were tested with the published data from the miscible displacement experiments on the two repacked soils with different aggregate sizes. Results show that both the inverse-gamma function and the CLT fit the measured breakthrough curves in the miscible displacement experiments. The estimates of the squared coefficient of variation of the pore-water velocity in the gamma pdf were 0.314 and 0.0582 for the two soils, and they were consistent with the lognormal pdf in the CLT.